Critical phenomena for systems under constraint
It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended t...
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irk-123456789-1528982019-06-14T01:26:24Z Critical phenomena for systems under constraint Izmailian, N. Kenna, R. It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality. Добре вiдомо, що накладання в’язей може змiнити критичнi властивостi системи. Раннi роботи Ессiма i Гарелiка, Фiшера та iн., присвяченi цьому явищу, зосереджувалися на впливi в’язей на головнi критичнi показники, якi описують фазовi переходи. Недавня робота розширила цi дослiдження на випадок критичних амплiтуд i показникiв для логарифмiчних поправок для деяких межових сценарiїв. Тут цi старi i новi результати зiбрано i пiдсумовано. Також обговорюється iнволютивна природа перетворень мiж критичними параметрами, якi описують iдеальну систему i систему з в’язями, при цьому особлива увага придiляється питанням, пов’язаним з унiверсальнiстю. 2014 Article Critical phenomena for systems under constraint / N. Izmailian, R. Kenna // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33602:1-9 — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 64.10.+h, 64.60.-i, 64.60.Bd DOI:10.5488/CMP.17.33602 arXiv:1406.3865 http://dspace.nbuv.gov.ua/handle/123456789/152898 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality. |
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Critical phenomena for systems under constraint |
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Critical phenomena for systems under constraint |
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Critical phenomena for systems under constraint |
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Critical phenomena for systems under constraint |
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critical phenomena for systems under constraint |
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Інститут фізики конденсованих систем НАН України |
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Critical phenomena for systems under constraint / N. Izmailian, R. Kenna // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33602:1-9 — Бібліогр.: 15 назв. — англ. |
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Condensed Matter Physics |
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2025-07-14T04:21:56Z |
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Condensed Matter Physics, 2014, Vol. 17, No 3, 33602: 1–9
DOI: 10.5488/CMP.16.33602
http://www.icmp.lviv.ua/journal
Critical phenomena for systems under constraint
N. Izmailian1,2, R. Kenna2
1 Yerevan Physics Institute, Alikhanian Brothers 2, 375036 Yerevan, Armenia
2 Applied Mathematics Research Center, Coventry University, Coventry CV1 5FB, England
Received June 5, 2014
It is well known that the imposition of a constraint can transform the properties of critical systems. Early work
on this phemomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the
leading critical exponents describing phase transitions. Recent work extended these considerations to critical
amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old
and new results are gathered and summarised. The involutory nature of transformations between the critical
parameters describing ideal and constrained systems are also discussed, paying particular attention to matters
relating to universality.
Key words: critical phenomena, Fisher renormalisation, universality
PACS: 64.10.+h, 64.60.-i, 64.60.Bd
1. Introduction
The study of thermodynamic systems subject to constraints has a long history. In 1966, Syozi and
Miyazima produced a diluted version of the Ising model and observed that annealed non-magnetic im-
purities affect the critical behaviour of the model [1]. In particular, the usual infinite critical peak in the
specific heat is replaced by a finite cusp. In 1967, Essam and Garelick quantified the nature of this change
as [2, 3]
αX =− α
1−α . (1.1)
Here, α represents the specific heat critical exponent for the ideal (non-diluted) system and αX is its
counterpart for the diluted system. If β and γ similarly represent the magnetisation and susceptibility
exponents, Essam and Garelick further showed that these transform to [2, 3]
βX = β
1−α , γX = γ
1−α . (1.2)
In 1968, Fisher produced a general theory for critical systems under constraint and the general process
linking the ideal critical exponents to those for the constrained system became known as Fisher renor-
malisation [4]. Because of their continued academic importance and relevance to real systems, phase
transitions in constrained systems remained a focus of study [5–9]. In recent years the transformation
has been extended to deal with other aspects of critical phenomena [10, 11].
Due to their experimental accessibility, amplitude terms are important for the description of criti-
cal phenomena. Unsurprisingly, these also change when a constraint is imposed. Perhaps surprisingly,
however, the precise nature of this transformation has only recently been studied [11]. Furthermore, in
certain marginal circmstances, multiplicative logarithmic corrections also enter the scaling description
at continuous phase transitions. Examples include those at the upper critical dimension of spin systems
and those at the border to regimes where the transition becomes first-order. The exponents of such loga-
rithmic corrections also transform when the system is subjected to a constraint [11].
© N. Izmailian, R. Kenna, 2014 33602-1
http://dx.doi.org/10.5488/CMP.16.33602
http://www.icmp.lviv.ua/journal
N. Izmailian, R. Kenna
To give a compact description of all these various aspects (leading critical exponents, logarithmic
corrections and amplitudes), we express the scaling behaviour of an ideal system as follows.
C (t ,0) = A±|t |−α| ln |t ||α̂ , (1.3)
m(t ,0) = B |t |β| ln |t ||β̂ for t < 0, (1.4)
χ(t ,0) = Γ±|t |−γ| ln |t ||γ̂ , (1.5)
m(0,h) = Dh
1
δ | ln |t ||δ̂ , (1.6)
ξ(t ,0) = N±|t |−ν| ln |t ||ν̂ . (1.7)
Here, t and h refer to the reduced temperature and magnetic field, respectively. The correlation length
in the absence an external field is ξ(t ,0). The subscripts + and − refer to amplitudes for t > 0 and t < 0,
respectively. In principle, we could employ subscripts for the critical exponents and their logarithmic
counterparts corresponding to those used for the amplitudes, but we suppress these here for simplicity
and because the exponents generally coincide on either side of the transition. Note that equation (1.3) for
the specific heat corresponds to an internal energy of the leading form
e(t ,0) =± A±
1−α |t |1−α| ln |t ||α̂. (1.8)
Finally, and for completeness, we mention that the leading form for the critical correlation function is as
follows:
G(t = 0,h = 0; x) = Θ
xd−2+η | ln x|η̂ . (1.9)
In what follows, we give a comprehensive overview of the effects of the presence of a constraint on
the critical exponents (including those of the logarithmic corrections, when present) and the amplitudes.
The critical exponents are universal quantities while the amplitudes are not. However, certain combi-
nations of amplitudes are universal. We show that the renormalisation process (Fisher renormalisation)
which transforms the universal critical paramenters is involutary in the sense that applying it twice re-
sults in the identity transformation. However, quantities which are not universal do not transform as
involutions. We also show that the various scaling relations between the critical parameters (exponents
and amplitudes) also hold for the transformed quantities.
In the next section, we summarise the scaling relations for the leading exponents, their logarithmic
counterparts and the universal amplitude combinations. In section 3 we apply the renormalisation pro-
cess and study its effects in section 4. We conclude in section 5.
2. Scaling relations and universal amplitude combinations
The four standard scaling relations are (see, e.g., [12] and references therein)
α+dν = 2, (2.1)
α+2β+γ = 2, (2.2)
(δ−1)β = γ , (2.3)
(2−η)ν = γ , (2.4)
where d represents the dimensionality of the system. The corresponding scaling relations for the loga-
rithmic-correction exponents are as follows:
α̂+d ν̂ = d ϙ̂ , (2.5)
α̂+ γ̂ = 2β̂ , (2.6)
(δ−1)β̂+ γ̂ = δδ̂ , (2.7)
(2−η)ν̂+ η̂ = γ̂ , (2.8)
33602-2
Critical phenomena for systems under constraint
where α̂ is augmented by unity in certain special circumstances described in [13]. The exponent ϙ̂
(“koppa-hat”) characterises the leading logarithmic correction to the finite-size scaling of the correlation
length ξL(0,0) ∼ L(lnL)ϙ̂, where L is the finite extent of the system [14]. It is the logarithmic counterpart
of the exponent ϙ, recently introduced to characterise the finite-size correlation length above the upper
critical dimension: ξL(0,0) ∼ Lϙ [14]. The relations (2.1)–(2.4) for the leading exponents are derived in the
Appendix, where it is also shown that they correspond to the following universal ratios [15]:
Rξ = A±N d
± , (2.9)
Rc = A±Γ±
B 2 , (2.10)
Rχ = Γ±Bδ−1
Dδ
, (2.11)
Q = ΘN 2−η
±
Γ±
. (2.12)
For the derivation of the logarithmic scaling relations (2.5)–(2.8), the reader is referred to [13]
In the next section, we examine the effects of constraints on the critical exponents and amplitudes. It
will turn out that the renormalised critical exponents obey the same set of scaling relations as their orig-
inal counterparts and that, when applied to universal quantities, Fisher renormalisation is involutory.
3. Fisher renormalisation
We consider a thermodynamic variable x conjugate to a field u, so that
x(t ,h,u) = ∂ fX (t ,h,u)
∂u
. (3.1)
Here, fX (t ,h,u) represents the free energy of the system under constraint and u represents a quantity
such as the chemical potential with x representing the density of annealed non-magnetic impurities. The
constraint is then expressed in terms of an analytic function as follows:
x(t ,h,u) = X (t ,h,u) . (3.2)
One may further assume that the singular part of the free energy of the constrained system is structured
analogously to its ideal counterpart f , so that
fX (t ,h,u) = f [t∗(t ,h,u),h∗(t ,h,u)] , (3.3)
up to a regular background term and in which t∗ and h∗
are analytic functions [4]. The ideal free energy
f (t ,h) is recovered if u is fixed at u = 0.
We assume that
h∗(t ,h,u) = hJ (t ,h,u) , (3.4)
so that h∗ = 0 when h = 0. Then,
∂h∗(t ,0,u)
∂t
= 0,
∂h∗(t ,0,u)
∂u
= 0, (3.5)
and
∂h∗(t ,h,u)
∂h
=J (t ,h,u)+h
∂J (t ,h,u)
∂h
, (3.6)
so that
∂h∗(t ,0,u)
∂h
=J (t ,0,u) . (3.7)
For simplicity, we also assume h →−h symmetry so that t∗ is a function of h2
. In that case,
∂t∗(t ,h,u)
∂h
∝ h , (3.8)
which vanishes at h = 0.
33602-3
N. Izmailian, R. Kenna
3.1. The critical point
To identify the critical point of the constrained system, one first writes the magnetization from equa-
tion (3.3) as follows:
mX (t ,h,u) = ∂ fX (t ,h,u)
∂h
= e(t∗,h∗)
∂t∗
∂h
+m(t∗,h∗)
∂h∗(t ,0,u)
∂h
. (3.9)
From equation (3.8), if the dependency on h is even, the first term on the right hand side of equation (3.9)
vanishes at h = 0. From equation (3.6), then
mX (t ,0,u) = m[t∗(t ,0,u),0]J (t ,0,u) . (3.10)
Now, the critical point of the ideal system is given by the vanishing ofm. Assuming thatJ (t ,0,u) is non-
vanishing, equation (3.10) gives thatmX (t ,0,u) vanishes only whenm[t∗(t ,0,u),0] = 0. This means that
critical point for the constrained system is given by
t∗(t ,0,u) = 0. (3.11)
(The vanishing of J (t ,0,u) would lead to two critical points instead of one for the constrained system.)
The Taylor expansion for the functionJ (t ,h,u) about the critical point is as follows:
J (t ,h,u) = J0 +b1t +·· ·+c1h +·· ·+c1(u −uc)+ . . . , (3.12)
where uc is the critical value of u for the constrained system. The critical point, therefore, has
J (0,0,uc) = J0.
3.2. The relation between t∗ and t
The constraint (3.2) determines the relation between t∗ and t . Equation (3.1) firstly gives
x(t ,h,u) = ∂ f (t∗,h∗)
∂t∗
∂t∗
∂u
+ ∂ fX (t∗,h∗)
∂h∗
∂h∗
∂u
. (3.13)
At h = 0, the second term on the right vanishes after equation (3.5). Therefore,
x(t ,0,u) = e(t∗,0)
∂t∗(t ,0,u)
∂u
. (3.14)
This will give a non-trivial relationship between t∗ and t . Expanding t∗(t ,0,u), one has
t∗(t ,0,u) = a1(u −uc)+ . . . , (3.15)
where uc and the coefficients of the expansion are non-universal. Therefore,
x(t ,0,u) = a1e(t∗,0)+ . . . , (3.16)
which, from equation (1.8), is as follows:
x(t ,0,u) =±a1
A±
1−α |t∗|1−α| ln |t∗||α̂+ . . . . (3.17)
On the other hand, Taylor expansion of the constraining function gives
X (t ,0,u) = X (0,0,uc)+d1(u −uc)+d2t + . . . (3.18)
= X (0,0,uc)+ d1
a1
t∗+d2t + . . . , (3.19)
from (3.15). Comparison with equation (3.16) leads to the vanishing of X (0,0,uc) and
±a1
A±
1−α |t∗|1−α| ln |t∗||α̂ = d1
a1
t∗+d2t + . . . . (3.20)
33602-4
Critical phenomena for systems under constraint
If α< 0, t∗ ∼ t and the renormalisation is trivial. In the case where α> 0, however, t renormalises to t∗
in a non-trivial manner. To describe this, define
a =
[
d2(1−α)
a1
] 1
1−α
. (3.21)
Then, the central result is that the constraint renormalises the reduced temperature from t to t∗, whereby
|t∗| = a
( |t |
A±
) 1
1−α | ln |t ||− α̂
1−α . (3.22)
3.3. Scaling for the constrained system
Equations (3.3), (3.5) and (3.22) deliver the leading internal energy and specific heat for the con-
strained system as follows:
eX (t ,0,u) = ∂ fX (t ,0,u)
∂t
= e(t∗,0)
∂t∗(t ,0,u)
∂t
=± a2−α
(1−α)2 A
−1
1−α
± |t | 1
1−α | ln |t ||− α̂
1−α , (3.23)
and
CX (t ,0,u) = ∂eX (t ,0,u)
∂t
= a2−α
(1−α)3 A
−1
1−α
± |t | α
1−α | ln |t ||− α̂
1−α , (3.24)
respectively. We identify the latter as follows:
CX (t ,0) = AX ±|t |−αX | ln |t ||α̂X , (3.25)
where
αX =− α
1−α , α̂X =− α̂
1−α , AX ± = a
1+ 1
1−αX (1−αX )3 AαX −1
± . (3.26)
The last relationship is non-universal since, besides A±, a is a non-universal constant.
The magnetization for the constrained system is given by equations (1.4), (3.10) and (3.12) as
mX (t ,0,u) = J0B |t∗|β| ln |t∗||β̂ for t < 0. In terms of t , we write
mX (t ,0) = BX |t |βX | ln |t ||β̂X for t < 0, (3.27)
and identify
βX = β
1−α , β̂X = β̂− βα̂
1−α , BX = J0aβ
B
AβX−
. (3.28)
Differentiating equation (3.9) with respect to h, delivers the susceptibility for the constrained sys-
tem and, using equation (3.8) at h = 0, together with equations (3.6) and (3.7), we obtain χX (t ,0,u) =
J 2
0χ(t∗,0) = ΓX ±|t |−γX | ln |t∗||γ̂X , or
χX (t ,0) = ΓX ±|t |−γX | ln |t ||γ̂X , (3.29)
where
γX = γ
1−α , γ̂X = γ̂+ γα̂
1−α , ΓX ± = J 2
0 a−γAγX
± Γ± . (3.30)
If δ> 1, the critical isotherm t = 0 has the leading magnetization in the field given by equations (3.6),
(3.8) and (3.9) asmX (0,h,u) = J0Dh
1
δ | lnh|δ̂. We identify
mX (0,h) = DX hδX | lnh|δ̂X , (3.31)
with
δX = δ , δ̂X = δ̂ , DX = J
1+ 1
δ
0 D. (3.32)
The critical exponents are, therefore, unchanged but the amplitude undergoes a transformation.
33602-5
N. Izmailian, R. Kenna
The correlation length renormalises in a similar way to the susceptibility since
ξX (t ) = ξ(t∗) = N±|t∗|−ν| ln |t∗||−ν̂.
We write
ξX (t ,0) = NX ±|t |−νX | ln |t ||−ν̂X , (3.33)
where
νX = ν
1−α , ν̂X = ν̂+ να̂
1−α , NX ± = a−νAνX
± N± . (3.34)
Finally, the correlation function is obtainable by differentiating the free energy with respect to two
local fields h1 = h(x1) and h2 = h(x2). One obtains
GX (t ,h,u; x) = ∂2 fX (t ,h,u)
∂h1∂h2
= J 2
0
∂2 f (t∗,h∗)
∂h∗
1∂h∗
2
= J 2
0G(t∗,h∗, x).
Setting t∗ = t = h∗ = h = 0, deliversGX (0,0,u; x) = J 2
0G(0,0, x). Writing
GX (0,0, x) = ΘX
xd−2+ηX
| ln x|η̂X , (3.35)
we identify
ηX = η , η̂X = η̂ , ΘX = J 2
0Θ. (3.36)
We have observed that neither the in-field magnetisation nor the correlation function exhibit non-
trivial renormalisation of the critical exponents. The former is the case by construction and the latter is
so because it is defined at the critical point. Likewise, the exponents ϙ and ϙ̂ governing finite-size scaling
of the correlaton length do not change under Fisher renormalisation, so that ϙX = ϙ and ϙ̂x = ϙ̂.
4. Properties of renormalised scaling parameters
It is straightforward to verify that if the critical exponents for the ideal system satisfy the scaling re-
lations (2.1)–(2.4), the renormalised exponents for the constrained system do likewise. (This observation
for the Essam-Fisher relation (2.2) was already made in [2].) The same statement applies to the scaling
relations for logarithmic corrections (2.5)–(2.8).
Fisher renormalisation applied to the universal critical exponents is involutory. This means that
renormalisation of renormalised exponents delivers pure values. For example, γX X = γX /(1−αX ) = γ
and γ̂X X = γ̂X +γX α̂X /(1−αX ) = γ̂. However, the same starement does not apply to the amplitudes. For
example, two successive applications of equation (3.30) give ΓX X ± different from Γ± .
Of course, the critical exponents, for which the transformation is involutory, are universal, whereas
the critical amplitudes are not. This observation prompts one to investigate the nature of the universal
combinations (2.9)–(2.12) under Fisher renormalisation. The non-universal terms J0 and a, which char-
acterise the transformations of the individual amplitude terms, drop out of the transformations of the
universal combinations through the scaling relations (2.1)–(2.4). The universal amplitude combinations
transform as follows:
RX c = 1
(1−α)3 Rc , (4.1)
RX χ = Rχ , (4.2)
RX ξ = 1
(1−α)3 Rξ , (4.3)
QX = Q , (4.4)
ZX = Z
U∆X
0
. (4.5)
Two successive applications of these transformations confirm the involutory nature of these universal
combinations.
33602-6
Critical phenomena for systems under constraint
5. Conclusions
Fisher renormalization, which generalises an earlier theory of Essam and Garelick is a staple of the
established theory of critical phenomena. The early work by these authors was extended in recent years
to encompass critical amplitudes and the exponents which govern logarithmic corrections to scaling,
when present. Here, a comprehensive treatment of all of these various elements has been given. We also
observe that the involutory nature of the renormalisation process is intrinsically linked to universality.
Acknowledgments
The work was supported by a Marie Curie IIF (Project no. 300206–RAVEN) and IRSES (Projects no.
295302–SPIDER and 612707–DIONICOS) within 7th European Community Framework Programme and by
the grant of the Science Committee of the Ministry of Science and Education of the Republic of Armenia
under contract 13–1C080.
A. Appendix: Universal amplitude Combinations
To identify the universal amplitude combinations, we begin with the standard scaling form for the
free energy and correlation length [12, 15]
f (t ,h) = b−d Y (Kt byt t ,Khbyh h) , (A.1)
ξ(t ,h) = bX (Kt byt t ,Khbyh h) . (A.2)
The scaling functions Y and X are universal and all the non-universality is contained in themetric factors
Kt and Kh .
Differentiating equation (A.1) with respect to h delivers the scaling form for the magnetization as
follows:
m(t ,h) = b−d+yh KhY (h)(Kt byt t ,Khbyh h) , (A.3)
where the parenthesized superscript signifies appropriate differentiaton of the scaling function. Setting
h = 0 and chosing
b = K
− 1
yt
t |t |− 1
yt (A.4)
gives the spontaneous magnetizationm(t ,0) = B(−t )β, for t < 0, in which
β= d − yh
yt
and B = K β
t KhY (h)(1,0) . (A.5)
On the other hand, setting t = 0 in equation (A.3) and choosing
b = K
− 1
yh
h h
− 1
yh , (A.6)
we obtainm(0,h) = Dh1/δ
in which
1
δ
= d − yh
yh
and D = K
1+ 1
δ
h Y (h)(0,1) . (A.7)
The susceptibility is obtained by differentiating equation (A.3) with respect to h. Again setting h = 0 and
using equation (A.4), one finds χ(t ,0) = Γ±|t |−γ, where
γ= 2yh −d
yt
and Γ± = K −γ
t K 2
h Y (hh)(±1,0) . (A.8)
For the specific heat, differentiate (A.1) twice with respect to t and again use equation (A.4) to find
C (t ,0) = A±|t |−α with
α= 2− d
yt
and A± = K 2−α
t Y (t t )(±1,0) . (A.9)
33602-7
N. Izmailian, R. Kenna
From equations (A.5) and (A.7), we can express yt and yh in terms of β and δ,
yt = d
β
1
δ+1
and yh = dδ
δ+1
. (A.10)
Similarly, using equations (A.5) and (A.7) we can express Kt and Kh in terms of B and D ,
Kt =
[
B
Y (h)(1,0)
] 1
β
[
D
Y (h)(0,1)
]− 1
β
δ
1+δ
and Kh =
[
D
Y (h)(0,1)
] δ
δ+1
. (A.11)
Here, the Y (h)
are universal while the amplitudes B and D are not.
Finally, expressing α and γ in terms of β and δ through equations (A.8) and (A.9), delivers the static
scaling relations (2.2) and (2.3). Correspondingly, one can express A± and Γ± in terms of B and D ,
Γ± = Y (hh)(±1,0)[
Y (h)(1,0)
] 1
β
[
Y (h)(0,1)
]δ B 1−δDδ , (A.12)
A± =
[
Y (h)(1,0)
]−(δ+1) [
Y (h)(0,1)
]δ
Y (t t )(±1,0)
Bδ+1
Dδ
. (A.13)
From the first of these, Γ±Bδ−1/Dδ
is a universal combination of universal factors. This is Rχ in equa-
tion (2.11). From the second, the ratio A±Dδ/Bδ+1
is universal. Or, combining with equation (A.7), the
quantity Rc in equation (2.10) is seen to be universal.
From equations (A.2) and (A.4), the correlation length is ξ(t ,0) = N±|t |−ν, where
ν= 1
yt
and N± = K
− 1
yt
t X (±1,0). (A.14)
From equation (A.9), the first of these is the hyperscaling relation (2.1). To connect N± to the other ampli-
tudes, one can exploit the relatonship between the susceptibility and the correlation function,
χ=
ξ∫
0
G(x)xd−1dx =Θξ2−η, (A.15)
from which Fisher’s scaling relation (2.4) follows, along with
Γ± =ΘN 2−η
± . (A.16)
The combination Q =ΘN 2−η
± /Γ± of equation (2.12) is, therefore, universal. Similarly, the universality of
Rξ in equation (2.9) can be explained through the hyperscaling relation f (t ,0) = A±|t |2−α/(2−α)(1−α) ∼
ξd (t ,0) = (N±|t |−ν)d
.
33602-8
Critical phenomena for systems under constraint
References
1. Syozi I., Miyazima S., Prog. Theor. Phys., 1966, 36, 1083; doi:10.1143/PTP.36.1083.
2. Essam J.W., Garelick H., Proc. Phys. Soc., 1967, 92, 136; doi:10.1088/0370-1328/92/1/320.
3. Garelick H., Essam J.W., J. Phys. C: Solid State Phys., 1968, 1, 1588; doi:10.1088/0022-3719/1/6/315.
4. Fisher M.E., Phys. Rev., 1968, 176, 257; doi:10.1103/PhysRev.176.257.
5. Lushnikov A.A., Phys. Lett. A, 1968, 27, 158; doi:10.1016/0375-9601(68)91180-8 [Sov. Phys. JETP, 1969, 29, 120].
6. Aharony A., J. Magn. Magn. Mater., 1978, 7, 215; doi:10.1016/0304-8853(78)90186-5.
7. Capel H.W., Perk J.H.H., den Ouden L.W.J., Phys. Lett. A, 1978, 66, 437; doi:10.1016/0375-9601(78)90388-2.
8. Capel H.W., den Ouden L.W.J., Perk J.H.H., Physica A, 1979, 95, 371; doi:10.1016/0378-4371(79)90024-4.
9. den Ouden L.W.J., Capel H.W., Perk J.H.H., Physica A, 1981, 105, 53; doi:10.1016/0378-4371(81)90063-7.
10. Kenna R., Hsu H.-P., von Ferber C., J. Stat. Mech., 2008, L10002; doi:10.1088/1742-5468/2008/10/L10002.
11. Izmailian N.Sh., R. Kenna, Preprint arXiv:1402.4673, 2014.
12. Fisher M.E., Rev. Mod. Phys., 1998, 70, 653; doi:10.1103/RevModPhys.70.653.
13. Kenna R., In: Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory, Yu. Holovatch
(Ed.), vol. 3, World Scientific, Singapore, 2012.
14. Kenna R., Berche B., Condens. Matter Phys., 2013, 16, 23601; doi:10.5488/CMP.16.23601.
15. Privman V., Hohenberg P.C., Aharony A., In: Phase Transitions and Critical Phenomena, vol. 14, Domb C.,
Lebowitz J.L. (Eds.), Academic, New York, 1991, pp. 1–134.
Критичнi явища для систем з в’язями
Н. Iзмаiлян1,2, Р. Кенна2
1 Єреванський фiзичний iнститут, м. Єреван, Вiрменiя
2 Центр прикладних математичних дослiджень, унiверситет м. Ковентрi, м. Ковентрi, Англiя
Добре вiдомо, що накладання в’язей може змiнити критичнi властивостi системи. Раннi роботи Ессiма i
Гарелiка, Фiшера та iн., присвяченi цьому явищу, зосереджувалися на впливi в’язей на головнi критичнi
показники, якi описують фазовi переходи. Недавня робота розширила цi дослiдження на випадок кри-
тичних амплiтуд i показникiв для логарифмiчних поправок для деяких межових сценарiїв. Тут цi старi
i новi результати зiбрано i пiдсумовано. Також обговорюється iнволютивна природа перетворень мiж
критичними параметрами, якi описують iдеальну систему i систему з в’язями, при цьому особлива увага
придiляється питанням, пов’язаним з унiверсальнiстю.
Ключовi слова: критичнi явища, ренормалiзацiя Фiшера, унiверсальнiсть
33602-9
http://dx.doi.org/10.1143/PTP.36.1083
http://dx.doi.org/10.1088/0370-1328/92/1/320
http://dx.doi.org/10.1088/0022-3719/1/6/315
http://dx.doi.org/10.1103/PhysRev.176.257
http://dx.doi.org/10.1016/0375-9601(68)91180-8
http://dx.doi.org/10.1016/0304-8853(78)90186-5
http://dx.doi.org/10.1016/0375-9601(78)90388-2
http://dx.doi.org/10.1016/0378-4371(79)90024-4
http://dx.doi.org/10.1016/0378-4371(81)90063-7
http://dx.doi.org/10.1088/1742-5468/2008/10/L10002
http://arxiv.org/abs/1402.4673
http://dx.doi.org/10.1103/RevModPhys.70.653
http://dx.doi.org/10.5488/CMP.16.23601
Introduction
Scaling relations and universal amplitude combinations
Fisher renormalisation
The critical point
The relation between t* and t
Scaling for the constrained system
Properties of renormalised scaling parameters
Conclusions
Appendix: Universal amplitude Combinations
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